Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠A1 is smaller than ∠A2.)
b = 47, c = 45, ∠C = 32°
∠A1 =
°
∠A2 =
°
∠B1 =
°
∠B2 =
°
a1 =
a2 =
First, we need to find the third angle ∠A by using the formula:
∠A = 180° - ∠C - ∠B
∠A = 180° - 32° - ∠B
Now, we can use the Law of Sines to find the possible values for angles and sides. The Law of Sines states:
a/sin∠A = b/sin∠B = c/sin∠C
1. Using the given information:
a/sin∠A = 45/sin∠C
a/sin∠A = 45/sin(32°)
a/sin∠A = 45/sin(32°)
a/sin∠A = 45/0.5299
a/sin∠A ≈ 84.9
2. Using the side b:
b/sin∠B = 45/sin∠C
47/sin∠B = 45/sin(32°)
47/sin∠B = 45/0.5299
47/sin∠B ≈ 51.4
Now, we can find the possible values for angles and sides:
∠A1 = sin^(-1)(84.9*sin(32°)/47) ≈ 57.0°
∠A2 = 180° - ∠A1 ≈ 123.0°
∠B1 = sin^(-1)(47*sin(32°)/45) ≈ 31.0°
∠B2 = 180° - ∠B1 ≈ 149.0°
a1 ≈ 84.9
a2 ≈ 84.9